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Lecture Notes on Stochastic Networks Lecture Notes on Stochastic Networks Frank Kelly and Elena Yudovina Contents Preface page ix Overview 1 Queueing and loss networks 2 Decentralized optimization 4 Random access networks 5 Broadband networks 6 Internet modelling 8 Part I 11 1 Markov chains 13 1.1 Definitions and notation 13 1.2 Time reversal 16 1.3 Erlang’s formula 18 1.4 Further reading 21 2 Queueing networks 22 2.1 An M/M/1 queue 22 2.2 A series of M/M/1 queues 24 2.3 Closed migration processes 26 2.4 Open migration processes 30 2.5 Little’s law 36 2.6 Linear migration processes 39 2.7 Generalizations 44 2.8 Further reading 48 3 Loss networks 49 3.1 Network model 49 3.2 Approximation procedure 51 3.3 Truncating reversible processes 52 v vi Contents 3.4 Maximum probability 57 3.5 A central limit theorem 61 3.6 Erlang fixed point 67 3.7 Diverse routing 71 3.8 Further reading 81 Part II 83 4 Decentralized optimization 85 4.1 An electrical network 86 4.2 Road traffic models 92 4.3 Optimization of queueing and loss networks 101 4.4 Further reading 107 5 Random access networks 108 5.1 The ALOHA protocol 109 5.2 Estimating backlog 115 5.3 Acknowledgement-based schemes 119 5.4 Distributed random access 125 5.5 Further reading 132 6 Effective bandwidth 133 6.1 Chernoff bound and Cramer’s´ theorem 134 6.2 Effective bandwidth 138 6.3 Large deviations for a queue with many sources 143 6.4 Further reading 148 Part III 149 7 Internet congestion control 151 7.1 Control of elastic network flows 151 7.2 Notions of fairness 158 7.3 A primal algorithm 162 7.4 Modelling TCP 166 7.5 What is being optimized? 170 7.6 A dual algorithm 172 7.7 Time delays 173 7.8 Modelling a switch 177 7.9 Further reading 185 8 Flow level Internet models 186 Contents vii 8.1 Evolution of flows 186 8.2 α-fair rate allocations 187 8.3 Stability of α-fair rate allocations 189 8.4 What can go wrong? 192 8.5 Linear network with proportional fairness 195 8.6 Further reading 199 Appendix A Continuous time Markov processes 201 Appendix B Little’s law 204 Appendix C Lagrange multipliers 207 Appendix D Foster–Lyapunov criteria 210 References 217 Index 221 Preface Communication networks underpin our modern world, and provide fasci- nating and challenging examples of large-scale stochastic systems. Ran- domness arises in communication systems at many levels: for example, the initiation and termination times of calls in a telephone network, or the sta- tistical structure of the arrival streams of packets at routers in the Internet. How can routing, flow control, and connection acceptance algorithms be designed to work well in uncertain and random environments? And can we design these algorithms using simple local rules so that they produce coherent and purposeful behaviour at the macroscopic level? The first two parts of the book will describe a variety of classical models that can be used to help understand the behaviour of large-scale stochastic networks. Queueing and loss networks will be studied, as well as random access schemes and the concept of an effective bandwidth. Parallels will be drawn with models from physics, and with models of traffic in road networks. The third part of the book will study more recently developed models of packet traffic and of congestion control algorithms in the Internet. This is an area of some practical importance, with network operators, content providers, hardware and software vendors, and regulators actively seeking ways of delivering new services reliably and effectively. The complex in- terplay between end-systems and the network has attracted the attention of economists as well as mathematicians and engineers. We describe enough of the technological background to communication networks to motivate our models, but no more. Some of the ideas described in the book are finding application in financial, energy, and economic net- works as computing and communication technologies transform these ar- eas. But communication networks currently provide the richest and best- developed area of application within which to present a connected account of the ideas. The lecture notes that have become this book were used for a Mas- ix x Preface ters course (“Part III”) in the Faculty of Mathematics at the University of Cambridge. This is a one-year postgraduate course which assumes a math- ematically mature audience, albeit from a variety of different mathemati- cal backgrounds. Familiarity with undergraduate courses on Optimization and Markov Chains is helpful, but not absolutely necessary. Appendices are provided on continuous time Markov processes, Little’s law, Lagrange multipliers, and Foster–Lyapunov criteria, reviewing the material needed. At Cambridge, students may attend other courses where topics touched on in these notes, for example Poisson point processes, large deviations, or game theory, are treated in more depth, and this course can serve as addi- tional motivation. Suggestions on further reading are given at the end of each chapter; these include reviews where the historical development and attribution of results can be found – we have not attempted this here – as well as some recent papers which give current research directions. The authors are grateful to students of the course at Cambridge (and, for one year, Stanford) for their questions and insights, and especially to Joe Hurd, Damon Wischik, Gaurav Raina, and Neil Walton, who produced earlier versions of these notes. Frank Kelly and Elena Yudovina Overview This book is about stochastic networks and their applications. Large-scale systems of interacting components have long been of interest to physicists. For example, the behaviour of the air in a room can be described at the mi- croscopic level in terms of the position and velocity of each molecule. At this level of detail a molecule’s velocity appears as a random process. Con- sistent with this detailed microscopic description of the system is macro- scopic behaviour best described by quantities such as temperature and pres- sure. Thus the pressure on the wall of the room is an average over an area and over time of many small momentum transfers as molecules bounce off the wall, and the relationship between temperature and pressure for a gas in a confined volume can be deduced from the microscopic behaviour of molecules. Economists, as well as physicists, are interested in large-scale systems, driven by the interactions of agents with preferences rather than inanimate particles. For example, from a market with many heterogeneous buyers and sellers there may emerge the notion of a price at which the market clears. Over the last 100 years, some of the most striking examples of large- scale systems have been technological in nature and constructed by us, from the telephony network through to the Internet. Can we relate the microscopic description of these systems in terms of calls or packets to macroscopic consequences such as blocking probabilities or throughput rates? And can we design the microscopic rules governing calls or packets to produce more desirable macroscopic behaviour? These are some of the questions we address in this book. We shall see that there are high-level constructs that parallel fundamental concepts from physics or economics such as energy or price, and which allow us to reason about the systems we design. In this chapter we briefly introduce some of the models that we shall en- counter. We’ll see in later chapters that for the systems we ourselves con- 1 2 Overview struct, we are sometimes able to use simple local rules to produce macro- scopic behaviour which appears coherent and purposeful. Queueing and loss networks We begin Chapter 1 with a brief introduction to Markov chains and Markov processes, which will be the underlying probabilistic model for most of the systems we consider. In Chapter 2 we study queueing networks, in which customers (or jobs or packets) wait for service by one or several servers. Figure 1 A network of four queues. First, we look at a single queue. We shall see how to model it as a Markov process, and derive information on the distribution of the queue size. We then look briefly at a network of queues. Starting from a simpli- fied description of each queue, we shall obtain information about the sys- tem behaviour. We define a traffic intensity for a simple queue, and identify Poisson flows in a network of queues. A natural queueing discipline is first-come-first-served, and we also look at processor sharing, where the server shares its effort equally over all the customers present in the queue. C=3 C=2 Figure 2 A loss network with some of the routes highlighted. In Chapter 3, we move on to consider loss networks. A loss network consists of several links, each of which may have a number of circuits. The classical example of its use is to model landline telephone connections Overview 3 between several cities. When a telephone call is placed between one node and another, it needs to hold simultaneously a circuit on each of the links on a route between the two nodes – otherwise, the call is lost. Note the key differences between this and a queueing network, which are summarized in Table 1. Table 1. Queueing versus loss networks Queueing networks Loss networks Sequential use of resources Simultaneous resource possession Congestion = delay Congestion = loss ⇒ ⇒ First, we treat loss networks where routing is fixed in advance. We show that the simple rules for call acceptance lead to a stationary distribution for the system state that is centred on the solution to a certain optimiza- tion problem, and we’ll relate this problem to a classical approximation procedure.
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